Motivation: theoretically relevant units \(\neq\) spatial units at which data are available

 

Example: data for different variables are available at different units

Outcome


Treatment


Instrument

 

 

Example: borders, number of units change over time

1937


1945


1991

 

 

Example: data are measured at different levels of geographic precision

admin 0


admin 1


admin 2

 

 

Example: different definitions of same units across data sources

admin 2


“admin 2”


admin 2

 

The dilemma for analysts

 

  1. Conduct analysis at theoretically inappropriate units

    • this is only possible if all data are available for those same units

 

   or

 

  1. Convert the data to a common set of (more appropriate) units

    • this is an intermediate, messy step
    • it always entails some information loss
    • it can lead to measurement error and biased estimation of quantities of interest
    • problem is well-known in geostatistics and social science
    • but no best practices exist for implementation, comparison, evaluation

Changes of support

Definitions

What are change of support problems?

 

  1. Geographic support: area, shape, size, and orientation associated with a variable’s spatial measurement
  2. Change of support (CoS) problem: making statistical inferences about a variable at one support by using data from a different support

 

Related topics:

  • ecological inference (EI): deducing micro variation from aggregate data
  • modifiable areal unit problem (MAUP): statistical inferences depend on the geographical regions at which data are observed

 

EI and MAUP are both special cases of CoS problems

The complexity of a CoS depends on

 

  1. Relative scale: aggregation, disaggregation, hybrid
  2. Relative nesting: whether one set of units falls completely, neatly inside other

Nesting and scale

Illustration

 

Let’s consider three sets of units (from the U.S. state of Georgia)

Example: different definitions of same units across data sources

precincts


constituencies


.5\(^\circ\) grid

  1. Suppose one wants to change the support from precincts to constituencies
    • scale: are source units smaller or larger than destination units?
    • nesting: do source units fit completely/neatly into destination units?

 

source units


source \(\cap\) destination


destination units

  1. Suppose one wants to change the support from constituencies to grid cells
    • scale: are source units smaller or larger than destination units?
    • nesting: do source units fit completely/neatly into destination units?

 

source units


source \(\cap\) destination


destination units

  1. Change of support #1 looks like an aggregation of nested units
  2. Change of support #2 looks like (mostly?) disaggregation of non-nested units

 

precinct \(\to\) constituency


constituency \(\to\) grid

Some considerations

 

  • many CoS problems require both aggregation and disaggregation
  • just because units are politically nested doesn’t mean they are geometrically nested (e.g. measurement error, imprecision of boundaries)
  • not always easy to “eyeball” these things
  • to get a better read on this, we need quantitative measures


Guesstimation ain’t easy

Formally

  • \(\mathcal{G}_S\): set of source polygons, indexed \(i=1,\dots,N_S\)
  • \(\mathcal{G}_D\): set of destination polygons, indexed \(j=1,\dots,N_D\)
  • \(\mathcal{G}_{S\cap D}\): intersection of \(\mathcal{G}_S\) & \(\mathcal{G}_D\), indexed \(i\cap j=1,\dots,N_{S\cap D}\)
  • \(a_i\): area of source polygon \(i\);\(\quad\) \(a_j\): area of destination polygon \(j\)
  • \(a_{i\cap j}\): area of intersection \(i\cap j\)

define relative scale as \(RS = \frac{1}{N_{S\cap D}}\sum_{i\cap j}^{N_{S\cap D}}1(a_i<a_j)\)

  • values of 1 \(=\) aggregation; values of 0 \(=\) disaggregation; 0-1 \(=\) hybrid

define relative nesting as \(RN = \frac{1}{N_S}\sum_{i}^{N_S} \sum_j^{N_D}\left(\frac{a_{i\cap j}}{a_i} \right)^2\)

  • values of 1 \(=\) full nesting; values of 0 \(=\) no nesting; 0-1 \(=\) partial nesting

 

Informally

  • relative scale:
    share of intersections where source units smaller than destination units
  • relative nesting:
    share of source units that cannot be split across destination units

Application of relative scale and nesting to Georgia data: any surprises here?

 

Relative scale

source \(\to\) destination (a) (b) (c)
(a) precincts – 1.00 1.00
(b) constituencies 0.00 – 0.12
(c) .5\(^\circ\) grid 0.00 0.89 –


Relative nesting

source \(\to\) destination (a) (b) (c)
(a) precincts – 0.98 0.92
(b) constituencies 0.01 – 0.29
(c) .5\(^\circ\) grid 0.05 0.54 –

 


(a)


(b)


(c)


Change of support algorithms

A CoS algorithm specifies a transformation between source and destination units

  • \(x\): is a variable being transformed from support \(\mathcal{G}_S\) to \(\mathcal{G}_D\)
  • \(x_{\mathcal{G}D}\):is true value of variable \(x\) in destination units \(\mathcal{G}_D\)
  • \(\widehat{x_{\mathcal{G}D}}^{(k)}=f_k(x_{\mathcal{G}S})\): estimated value of \(x_{\mathcal{G}D}\), calculated w/ CoS algorithm \(k\)

these range from simple geometric operations to complex model-based predictions

 




Types of variables

  1. Extensive (depend on area and scale)

    • aggregates are (weighted) sums
    • must satisfy the pycnophylactic (mass-preserving) property:
      • if area is split or combined, its values must be split or combined
      • sum of values in destination units must equal sum in source units
    • examples: population counts, event counts, acreage, mineral deposits

  2. Intensive (don’t depend on area and scale)

    • aggregates are (weighted) means
    • examples: population density, vote margins, median income
    • intensive variables are often functions of extensive variables (density \(=\) mass/vol.)
    • best practice: reconstruct in destination units from transformed components
      (\(\widehat{\text{mass}}_{\mathcal{G}D}/\widehat{\text{volume}}_{\mathcal{G}D} = \widehat{\text{density}}_{\mathcal{G}D}\))


 

 

Examples

Areal interpolation

Areal weighting is the default CoS method in many commercial and open-source GIS

  1. Advantages

    • easy to implement
    • requires information only on geometry of source and destination units
    • no need for ancillary data

  2. Disadvantages

    • assumes that the phenomenon of interest is uniformly distributed in source units
    • this becomes less problematic if source units are relatively small
    • but more problematic as source units increase in size


Overlapping areas

Pseudocode for areal interpolation

  1. Intersect \(\mathcal{G}_{S}\) and \(\mathcal{G}_{D}\), creating a third polygon layer \(\mathcal{G}_{S\cap D}\),
    • each feature \(i\cap j\in \{1,\dots,N_{S\cap D}\}\) is a part of source polygon \(i\) that falls inside destination polygon \(j\).
  2. Compute area weights for each intersection \(i\cap j\), proportional to
    1. for extensive variables: \(w_{i\cap j}^{\text{(ext)}}=\frac{a_{i\cap j}}{a_i}\)
      (i.e. share of \(i\)’s area represented by intersection \(i\cap j\))
    2. for intensive variables: \(w_{i\cap j}^{\text{(int)}}=\frac{a_{i\cap j}}{a_j}\)
      (i.e. share of \(j\)’s area contributed by intersection \(i\cap j\))
  3. Combine weighted statistics for each destination polygon \(j\):
    1. \(\hat{x}_j=\sum_{i\cap j}^{N_{\cap j}} w_{i\cap j}x_{i\cap j}\), where \(x_{i\cap j}\) is the value of \(x\) in intersection \(i\cap j\) and \(N_{\cap j}\) is the number of intersections in \(j\)

Areal interpolation is just one of many potential CoS methods

 

Examples:

  • simple overlay
  • population weighted interpolation
  • ordinary kriging
  • universal kriging
  • thin-plate splines and random forests

these differ in their assumptions
(e.g. uniformity vs. heterogeneity) and requirements (e.g. ancillary data)

 

… what’s more important is not the choice of CoS algorithm, but the relative scale and nesting of source and destination units


Choice paralysis

Assessing transformation quality

Precinct-to-constituency CoS (\(RS=1, RN=0.98\))

 

Different CoS algorithms \(\to\) Different transformed values

–>

Constituency-to-grid CoS (\(RS=0.12, RN=0.29\))

 

But how do \(RS\), \(RN\) affect the quality of transformations (prediction error, rank correlation, estimation bias), holding CoS algorithm constant?

Higher \(RS\), \(RN\) \(\to\) Lower prediction error relative to true values

 

How RN and RS affect root mean squared error

Higher \(RS\), \(RN\) \(\to\) Higher correlation b/w transformed values & true values

 

How RN and RS affect correlation

Higher \(RS\), \(RN\) \(\to\) Less bias in regression coefficients

 

How RN and RS affect OLS estimation bias

What is to be done?

  1. General recommendations:
    • consider relative scale and nesting as ex ante measures of CoS complexity
    • check face validity of transformed values through visualization
  2. If “ground truth” data (micro data, cross-unit IDs) are available:
    • validate transformed values with micro data
    • use micro data as source units
    • match on common ID (if units are well-nested)
  3. If “ground truth” data are not available:
    • be transparent about limitations/assumptions
    • partial validation (if micro data available for some regions)
    • report results from alternative CoS algorithms when possible

 

Bad news: \(RN\) and \(RS\) can be calculated in R (SUNGEO::nesting()), not QGIS
(but you can still do CoS in QGIS, using good judgement and common sense!)